21. Solution: The possible values of $X$ are $-1, 0, 1$.
$$\begin{aligned}
& P(X = -1) = (1-\alpha)\beta, \\
& P(X = 0) = \alpha\beta + (1-\alpha)(1-\beta), \\
& P(X = 1) = \alpha(1-\beta),
\end{aligned}$$
Therefore the distribution table of $X$ is
| $X$ | $-1$ | $0$ | $1$ |
| $P$ | $(1-\alpha)\beta$ | $\alpha\beta + (1-\alpha)(1-\beta)$ | $\alpha(1-\beta)$ |
(2) (i) From (1) we have $a = 0.4$, $b = 0.5$, $c = 0.1$. Therefore $p_i = 0.4p_{i-1} + 0.5p_i + 0.1p_{i+1}$, so $0.1(p_{i+1} - p_i) = 0.4(p_i - p_{i-1})$, that is $p_{i+1} - p_i = 4(p_i - p_{i-1})$. Since $p_1 - p_0 = p_1 \neq 0$, the sequence $\{p_{i+1} - p_i\}$ $(i = 0, 1, 2, \cdots, 7)$ is a geometric sequence with common ratio 4 and first term $p_1$.
(ii) From (i) we obtain $p_8 = p_8 - p_7 + p_7 - p_6 + \cdots + p_1 - p_0 + p_0 = (p_8 - p_7) + (p_7 - p_6) + \cdots + (p_1 - p_0) = \frac{4^8 - 1}{3}p_1$. Since $p_8 = 1$, we have $p_1 = \frac{3}{4^8 - 1}$, therefore $p_4 = (p_4 - p_3) + (p_3 - p_2) + (p_2 - p_1) + (p_1 - p_0) = \frac{4^4 - 1}{3}p_1 = \frac{1}{257}$. $p_4$ represents the probability of ultimately concluding that drug A is more effective. From the calculation result, we can see that when drug A has a cure rate of 0.5 and drug B has a cure rate of 0.8, the probability of concluding that drug A is more effective is $p_4 = \frac{1}{257} \approx 0.0039$. The probability of reaching an incorrect conclusion is very small, indicating that this experimental scheme is reasonable.